An illusionist lays a full deck of 52 cards in a row and announces that, after 51 removals, only the Three of Clubs will remain. At each step a spectator names a valid position number. The illusionist chooses whether to count that position from the left or from the right and removes the corresponding card. For which initial positions of the Three of Clubs can the illusionist guarantee success?
Let $\omega$ be a circle with centre $O$ and distinct points $A,C$ on it. For any point $P$ on $\omega$ distinct from $A,C$, let $X,Y$ be the midpoints of $AP,CP$. Let $H$ be the orthocentre of triangle $OXY$. Prove that the position of $H$ does not depend on the choice of $P$.
A row of 100 squares contains one counter in each square. Swapping counters in adjacent squares costs 1 dollar. Swapping two counters with exactly three counters between them is free. What is the least amount of money needed to reverse their order? Give the number of dollars.
Let $a_1,\ldots,a_{1000}$ be integers and write their squares around a circle. The sum of any 41 consecutive numbers on this circle is divisible by $41^2$. Must each $a_i$ be divisible by 41?
Vasya (called Basil in the supplied Vietnamese text) has unlimited $1\times1\times3$ bricks and L-shaped bricks, each made from three unit cubes. He has completely filled an $m\times n\times k$ box with these bricks, where $m,n,k$ are integers greater than 1. Prove that the same box can be filled using only L-shaped bricks.
